On (local) metric dimension of graphs with m-pendant points

Abstract

An ordered set of vertices S is called as a (local) resolving set of a connected graph ),( EVG if for any two adjacent vertices GG Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 Vts have distinct representation G with respect to S, that is ).|()|( StrSsr A representation of a vertex in G is a vector of distances to vertices in S. The minimum (local) resolving set for G is called as a (local) basis of G. A (local) metric dimension for G denoted by dim(G), is the cardinality of vertices in a basis for G, and its local variant by dim (G). Given two graphs, G with vertices s l 1 , s 2 , ..., s p and edges e 1 , e 2 , ..., e , and H. An edgecorona of G and H, GH is defined as a graph obtained

by taking a copy of G and q copies of

H

and for each edge e j = s i s h of G joining edges between the two end-vertices s q and each vertex of j-copy of H. In this paper, we determine and compare between the metric dimension of graphs with m pendant

points, GmK , and its local variant for any connected graph G. We give an upper bound of the dimensions.